Programme

The conference runs from the morning of Tuesday 25 June to the early afternoon on 28 June. Since it can take a little bit longer to reach Oulu, we recommend arriving on Monday 24 June.

Schedule

An indicative time table can be found below. Please note that is still subject to change. All of the talks will take place at lecture hall L2 on the University of Oulu Linnanmaa campus.

Titles & Abstracts

Tuesday

• 09:30 – 10:30: Pablo Shmerkin (University of British Columbia)
  • Title: Distances, incidences, projections of Ahlfors regular sets
  • Abstract: In the last decade it was realized that many classical problems in geometric measure theory are more tractable for Ahlfors regular sets (perhaps in an approximate or finitary sense) than for general sets. I will survey some recent progress in this direction, obtained in (separate) joint work with H. Wang and with T. Orponen. The main goal of the talk will be to indicate how the Ahlfors regularity assumption comes into play.
• 11:00 – 12:00: Balázs Bárány (Budapest University of Technology and Economics)
  • Title: A simultaneous dimension result via transversality
  • Abstract: To study the dimension theory of general self-conformal IFSs with overlaps, a widely used method is to study parametrized families of such IFSs instead of individual ones. If such a family satisfies the so-called transversality condition, then it has been known for two decades that for every invariant ergodic measure defined on the symbolic space, there is a full-measure subset of parameters for which the Hausdorff dimension of the push-forward measure can be determined by the ratio entropy over Lyapunov exponent. In this talk, we show that we can choose a full-measure subset of parameters for which the assertion above holds for all ergodic measures simultaneously. We apply this result to the multifractal analysis of typical self-similar measures with overlaps. This is a joint work with Károly Simon and Adam Śpiewak.
• 12:00 – 12:30: Efstathios Konstantinos Chrontsios Garitsis (University of Tennessee, Knoxville)
  • Title: Sharp bounds on the Assouad spectrum of graphs
  • Abstract: A sharp upper bound for the box-counting dimension of graphs of Hölder functions has been known since the introduction of the notion. On the other hand, precisely determining the dimension of the graph of a specific function has been challenging. The Hausdorff dimension of the Weierstrass function graph $\mathrm{graph}(W)$ was determined by Shen in 2015, while determining the Assouad dimension of $\mathrm{graph}(W)$ is still an open problem. In this talk, we focus on the collection of dimensions interpolating between the upper box-counting dimension and the Assouad dimension, namely the Assouad spectrum. We provide sharp upper bounds for the Assouad spectrum $\dim_A^\theta \mathrm{graph}(f)$ of the graph of real-valued Hölder and Sobolev functions $f$. In the setting of Hölder graphs, we further provide a geometric algorithm which takes as input the graph of an $\alpha$-Hölder continuous function satisfying a matching lower oscillation condition and returns the graph of a new $\alpha$-Hölder continuous function for which the Assouad $\theta$-spectrum realizes the stated upper bound for all $\theta\in (0,1)$. Examples of functions to which this algorithm applies include the functions of Weierstrass and Takagi. This talk is based on joint work with J. Tyson.
• 14:00 – 15:00: Jeff Steif (Chalmers University of Technology)
  • Title: Poisson Representable Processes
  • Abstract: Motivated by Alain-Sol Sznitman’s random interlacement process, we consider a general class of processes which can be constructed in a similar manner. Namely, one considers a general Poisson process on the collection of subsets of a given set S and the corresponding “Poisson Generated Process” is obtained by taking the union of the sets which arise in the Poisson process. In this way, we obtain a random subset of S or equivalently a 0-1 valued process indexed by S.
    Our main focus is to determine which processes are representable in this way. Some of our results are as follows. (1) All positively associated Markov chains and a large class of positively associated renewal processes are representable. (2) Whether an average of two product measures, with close densities, on $n$ variables, is representable is related to the zeroes of the polylogarithm functions. (3) Using (2), we show that a number of tree indexed Markov chains as well as the Ising model on $\mathbb{Z}^d$ for $d$ at least 2 for certain interaction parameters, is not representable. (4) The collection of permutation invariant processes which are representable corresponds exactly to the set of infinitely divisible random variables on $[0,\infty]$ via a certain transformation. (5) The supercritical Curie-Weiss model is not representable for large $n$.
    The talk is based on joint work with Malin P. Forsström and Nina Gantert.
• 15:00 – 15:30: Jörg Thuswaldner (University of Leoben)
  • Title: Metric removability, permeability, and dimension
  • Abstract: A set $\Theta\subset \mathbb{R}^d$ is called metrically removable if any two points $x,y\in \mathbb{R}^d$ can be connected by a path that is not much longer than the straight line connecting $x$ and $y$ and that intersects $\Theta$ at most in its end points. In this talk we provide results on various kinds of dimensions of metrically removable sets. We apply these results to exhibit classes of metrically removable self-similar sets. We also study permeability, which is a weaker variant of metric removability .(This is joint work with G.~Leobacher, A.~Steinicke, and T.~Rajala.)
• 16:00 – 17:00: Henna Koivusalo (University of Bristol)
  • Title: Classification of those cut and project sets which are substitution tilings
  • Abstract: Cut and project sets are obtained by taking an irrational slice through a lattice and projecting it to a lower dimensional subspace. This usually results in a set which has no translational period, even though it retains a lot of the regularity of the lattice. As such, cut and project sets are one of the archetypical examples of point sets featuring aperiodic order. The other canonical method for generating aperiodic order is via substitutions; point sets generated from an inflation-redecoration process. Many classical examples of aperiodic order, e.g. the famous Penrose tiling, have a description from both methods. It is hence fundamental to the theory of aperiodically ordered point sets to answer the question:
    What property characterises those cut and project sets which have a substitution description?
    In this talk I will give an overview of the definition and basic properties of cut and project sets, discuss classical examples of aperiodic order, define the notion of `pattern with a substitution rule’, and finally come to a new result of mine, answering the question above. This work is joint with Jamie Walton (Nottingham) and Edmund Harriss (Arkansas).
• 17:00 – 18:00: Han Yu (Warwick University)
  • Title: Multiplicative approximation: Littlewood, Gallagher, and beyond
  • Abstract: A classical conjecture of Littlewood asks whether all irrational pairs of reals $(x,y)$ satisfy $\liminf_{n\to\infty} n||nx||||ny||=0$. A classical result of Gallagher says that this conjecture is true (with some logarithmic buffers) for Lebesgue almost all pairs. In this talk, we are looking at replacing the Lebesgue measure with some other measures, e.g. on manifolds for fractals. We will survey some open problems as well as some recent results. (Joint work with Sam Chow)

Wednesday

• 09:30 – 10:30: De-Jun Feng (Chinese University of Hong Kong)
  • Title: Homogeneous iterated function systems with the weak separation condition
  • Abstract: In this talk, I will present some partial results on the characterisation of homogeneous iterated function systems on the line which have the attractor [0,1] and satisfy the weak separation condition. It is based on joint work with Ching-Yin Chan.
• 11:00 – 12:00: Eveliina Peltola (Aalto University)
  • Title: On Loewner evolutions with jumps
  • Abstract: I discuss the behavior of Loewner evolutions driven by Levy processes. Schramm’s celebrated version (Schramm-Loewner evolution), driven by standard Brownian motion, has been a great success for describing critical interfaces in statistical physics for the past 25 years. Loewner evolutions with other random drivers have been proposed, for instance, as candidates for finding extremal multifractal spectra (towards Brennan’s conjecture and related problems in function theory), and for generating some tree-like growth processes in statistical physics (e.g., non-self-similar growth, versions of DLA, etc.).
    However, very few basic questions (and only in special cases) regarding the geometric behavior of such Loewner evolutions, e.g., whether they can ge generated by curves, whether the hulls are locally connected, tree-like, or forest-like, have been answered thus far. In this talk, I pertain to the case of general Levy drivers and describe results and ongoing work on the geometric behavior of Loewner evolutions with jumps.
    Based on joint work with Anne Schreuder (Cambridge).
• 12:00 – 12:30: Amlan Banaji (University of Loughborough)
  • Title: Lower box dimension of infinitely generated self-conformal sets
  • Abstract: Consider the following natural question: for which classes of fractal sets does the box dimension exist? After surveying some known results, I will discuss recent joint work with Alex Rutar where we show that non-existence of box dimension is possible even within the well-studied class of numbers whose continued fraction expansion consists of digits restricted to a given infinite subset of the natural numbers. This result is in fact a consequence of a formula which we prove for the lower box dimension of infinitely generated self-conformal sets, combined with Mauldin and Urbanski’s formula for the upper box dimension of such sets from the 1990s. Interestingly, the lower box dimension (unlike upper box dimension) depends on more refined scaling properties of the set F of fixed points of the contraction maps than just the lower and upper box dimensions of F.
• 14:00 – 15:00: Sylvester Eriksson-Bique (University of Jyväskylä)
  • Title: Resolution of Kleiner’s conjecture
  • Abstract: Kleiner posed the following conjecture in his 2006 ICM talk: every self-similar and combinatorially Loewner space is quasisymmetric to a Loewner space. Firstly, everything that is quasisymmetric to Loewner is combinatorially Loewner. Thus, this would be a converse to this statement. It would have been a very interesting converse, since as Bourdon and Kleiner observed, there are a lot of spaces with the combinatorial Loewner property: basically almost anything with enough symmetry. This includes the Sierpinski carpet, Menger curve, boundaries of many hyperbolic groups… It would be remarkable, if all of these spaces were quasisymmetric to Loewner spaces. We disprove Kleiner’s conjecture by constructing a large family of explicit counterexamples. These examples arise from a new construction of fractals by means of a linear replacement rule, and the proof that they are counterexamples is a pretty application of porosity and explicit modulus computations. These spaces resemble Laakso-spaces, but expand on them by allowing ``removable edges’’. This class of spaces is remarkable in that one can efficiently and in closed form compute optimizers for potential problems – and thus study effectively analysis on fractals. Despite the very natural construction, these fractals have been missed in the existing literature – both on the random walk side and quasiconformal side. I will also discuss how Kleiner’s conjecture is related to the existence of analytically one-dimensional spheres. This is joint with with Riku Anttila and Guy C. David.

Thursday

• 09:30 – 10:30: Tuomas Orponen (University of Jyväskylä)
  • Title: Escaping from fractals along straight lines
  • Abstract: Nikodym in 1927 constructed a full measure Borel subset $N$ of the unit square such that one can escape from every point $x$ in $N$ along a straight line: in other words, there exists a straight line intersecting $N$ only at $x$. Can there be many such lines? I will discuss a result saying that if one can escape from every point of $N$ along an $s$-dimensional set of lines, then the Hausdorff dimension of $N$ cannot exceed $2 - s$. Based on joint work with Damian Dabrowski and Max Goering.
• 11:00 – 12:00: Xiong Jin (Manchester University)
  • Title: On dimension conservation of self-similar sets with dense rotations
  • Abstract: I will talk about the dimension conservation property of self-similar sets, that is the dimension of the projections plus the dimension of fibres should be close to the dimension of the self-similar sets. Firstly I will present a review of the currents known results in the planar case and then I will report some new results in the higher dimensional case.
• 12:00 – 12:30: Pieter Allaart (University of North Texas)
  • Title: Random subsets of Cantor sets generated by trees of coin flips
  • Abstract: I will introduce a natural way to construct a random subset of a homogeneous two-part Cantor set $C$ via random labelings of an infinite binary tree. Each edge in the tree is independently labeled $0$ with probability $p$, or $1$ with probability $1-p$. Each infinite labeled path down the tree thus determines a unique point in $C$, and the collective of all these points is our random subset. When $p=1/2$, the random subset is almost surely of full Hausdorff dimension in $C$ but of zero Hausdorff measure. For $p\neq 1/2$, the dimension of $F$ is strictly smaller than that of $C$; I will present non-trivial upper and lower bounds for the Hausdorff dimension. The results may be generalized to iterated function systems with more maps and trees with more branches. Time permitting, I will also discuss what happens when we label the edges of the tree deterministically, in a periodic fashion. Based on joint work with Taylor Jones.
• 14:00 – 15:00: Kalle Kytölä (Aalto University)
  • Title: Boundary visits of SLE and lattice model interfaces
  • Abstract: This talk addresses two related topics: boundary visits of SLE random curves and of interfaces in critical lattice models. We review some results (of Alberts, Sheffield, Lawler, and others) about the covariant measure of SLEs on the boundary, which is supported on the fractal set of boundary visit points of the curve. We then present a conjecture based on conformal field theory and quantum group method about exact multi-point correlations of both this measure and scaling limits of lattice model interface boundary visit probabilities, and we compare it with numerical simulations of lattice models. In one case the conjecture is now a theorem: renormalized scaling limits of arbitrary ordered n-point boundary visit probabilities of a branch of a uniform spanning tree (or equivalently of a loop-erased random walk) have been proven to exist and to satisfy the predicted combined system of second and third order partial differential equations and boundary conditions.
• 15:00 – 15:30: Tamas Keleti (Eötvös Loránd University)
  • Title: New Hausdorff type dimensions and optimal bounds for bilipschitz invariant dimensions
  • Abstract: We introduce a new family of fractal dimensions by restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension. Among others, it turns out that this family contains continuum many distinct dimensions, and they share most of the properties of the Hausdorff dimension, which answers negatively a question of Fraser. We also consider the supremum of these new dimensions, which turns out to be another interesting notion of fractal dimension.
    We prove that among those bilipschitz invariant, monotone dimensions on the compact subsets of $\mathbb{R}^n$ that agree with the similarity dimension for the simplest self-similar sets, the modified lower dimension is the smallest and when $n=1$ the Assouad dimension is the greatest, and this latter statement is false for n>1. This answers a question of Rutar.
    Joint work with Richárd Balka.
• 16:00 – 17:00: Steffen Winter (Karlsruhe Institute of Technology)
  • Title: Support measures and complex dimensions
  • Abstract: For any given closed set in $\mathbb{R}^d$, the general Steiner formula of Hug, Last and Weil describes its tube volume and the support measures arising from this formula encode its geometric properties. However, it is not easy to extract fractal properties of a set directly from these measures. We introduce some new geometric functionals for compact subsets of R^d which we call basic functionals and support functionals, which are tightly related to the support measures. We connect appropriate scaling exponents of these functionals to the (upper) Minkowski dimension of the set and explain how its Minkowski content may be represented in terms of these functionals.
    Our original motivation was to elucidate the geometric meaning of the coefficients in fractal tube formulas that arise in the theory of complex dimensions by Lapidus, Zubrinic and Radunovic. We explain the relations and also comment on the connections to fractal curvatures.
    Based on joint work with Goran Radunovic.

Lightning Talks & Posters

• Laura Breitkopf, Poster: Equidistribution of cusp points of Hecke triangle groups
• Attila Gáspár, Lightning talk: 1-Lipschitz maps onto polygons
• Nero Ziyu Li, Lighning talk: Fractal dimension for iterated graph systems
• Ridip Medhi, Poster: Ergodic Theorem for Iterated Function System with Local Radial Contraction
• Markus Myllyoja, Lightning talk: Hausdorff dimension of random covering sets generated by balls
• Ana de Orellana, Lightning talk: Exceptional projections and dimension interpolation
• Vilma Orgoványi, Lightning talk, Poster: Properties of some random self-similar sets with overlaps
• Quentin Rible, Lightning talk, Poster: Regularity study of function in inhomogeneous Besov Spaces and the Riemann’s series
• Alex Rutar, Lightning talk: Dynamical covering arguments and Assouad spectra of Gatzouras–Lalley carpets
• Lauritz Streck, Lightning talk: Fourier decay for inhomogeneous self-similar measures with rational contractions
• Yun Sun, Lightning talk, Poster: Topological expansive Lorenz maps with a hole at critical point
• Yu-Liang Wu, Poster: Thermodynamic formalisms for Markov subshifts on $d$-trees
• Yuhao Xie, Lightning talk: Dimensions of projections of typical self-affine sets.
• Tianhan Yi, Lightning talk: Homogeneous non-Rajchman self-similar measures on the line
• Xintian Zhang, Lighning talk: Hausdorff dimension of the set of $\psi$-badly approximable points

Friday

• 09:30 – 10:30: Hong Wang (NYU Courant Institute)
  • Title: Some structure of Kakeya sets in $\mathbb{R}^3$
  • Abstract: A Kakeya set in $\mathbb{R}^n$ is a set of points that contains a unit line segment in every direction. We study the structure of Kakeya sets in $\mathbb{R}^3$ and show that for any Kakeya set $K$, there exists well-separated scales $0<\delta<\rho\leq 1$ so that the $\delta$-neighborhood of $K$ is almost as large as the $\rho$-neighborhood of $K$. As a consequence, every Kakeya set in $\mathbb{R}^3$ has Assouad dimension $3$. This is joint work with Josh Zahl.
• 11:00 – 12:00: András Máthé (University of Warwick)
  • Title: Sum-product problem for the dimension of fractal sets with information inequalities
  • Abstract: Given a finite set of integers A, either the sumset A+A or the product set AA is significantly larger than the size of A, by a theorem of Erdos and Szemeredi. (Their open conjecture is that the number of elements in one of these sets is necessarily nearly the square of the number of elements of A.) Katz and Tao studied the discretised version of the fractal analogue of this problem, where A is a subset of the reals and its size is measured by its dimension. By a result of Bourgain, indeed at least one of A+A or AA has a strictly larger “dimension” than that of A. There have been many quantitative improvements on this estimate since.
    I will review this problem and some related questions in fractal geometry, and talk about how Shannon-type information inequalities can be used to obtain some good estimates. Based on joint work with Will O’Regan.
• 12:00 – 12:30: Aleksi Pyörälä (University of Jyväskylä)
  • Title: Summing a subset of the parabola with itself
  • Abstract: Given a set $A$ on the plane, the sumset $A+A = \{a+b: a,b \in A\}$ should in general be substantially larger than the set $A$, subsequent summations $A+A+…+A$ even more so. For finite sets, one can expect to discover such a phenomenon between the cardinalities of $A$ and $A+A+…+A$, and if $A$ is infinite, between the Hausdorff dimensions of $A$ and $A+A+…+A$. Based on a joint work with Tuomas Orponen and Carmelo Puliatti, I will discuss this phenomenon for subsets of the plane that lie on the parabola
    $\{(x,x^2): x \in \mathbb{R}\}$: If $A$ is a subset of the parabola and $s$ denotes the Hausdorff dimension of $A$, then for any $t < \min\{s+1, 3s\}$, the Hausdorff dimension of the $n$-fold sum $nA = A+A+…+A$ is larger than $t$ for all large enough $n$. With a similar method, we demonstrate the non-additivity of the parabola even further by establishing similar sharp bounds for Lp-norms of Fourier transforms of fractal measures supported on the parabola.